I MADE THIS BUILD BEFORE BUT I FORGOT OF LOGIN, SORRY.
Play with Corki has been a wonderfull jorney for me,he is a simple hero to control at same time he is dificult to play in right way(for me).So i make this guide to explain that if you wanna make this build for your Corki you first have to change the way you play with him.The essence of my build is:farming.This is the most important element to make this build works.To farming you have to use the most you can your gatling, your ult of course and if you want phosphorus bomb(i prefer gatling gun).This build have some items and spell to help you to do that like: clarity,madred and of couse your damage.I see and play a lot of games with Corki and almost all of then the Corki have only force to battle from distance of his oponent if his oponent's is strong in combat like:Jax,Trindamere,Xin Zhao,etc...
Then i have make some changes in Corki build in order to make Corki not only fight from distance but fight from all the ways(close or long distance of enemy).This changes are a little bit of armor,dogde(with first skill the oponent will miss a lot of attacks)and more damage that you can image(463 AD and with black cleaver and gatling gun the enemy have no armor against you).
This little changes make a lot of diference when you are one-on-one fights.I'm not saying that you could kill a jax if you are in same lvl and same money that him.I'm saying that if you use your abilitys and your critical bonus against minions that madred gives to you and farming money and lvl you could defeat jax.So solo mid is the best way of farming and get lvl with Corki.

In mine rune,i try to get the most attack speed i could.You will start the game with 28% more attack speed.I prefer focus mine runes on 2 elements to get more of each one(dogde,attack speed).4,5 dogde from runes.
Armor runes,armor penetration runes,damage runes etc...All this is good for Corki but as i say i prefer to focus only on elements thats for me are the essentials .

In the masteries of this build,as any other build masteries i focus in interact the strenghs of the this hero with his masteries.Example: everytime that Corki dogdes an attack he increase his movement speed in 10% until 10 seconds after the attack.Another example: his clarity gives nearby champions friends the same amount of mana that Corki receives,Corki's masteries also gives him armor what fits well with this build.This is a good masteries build but with you think that you can make better, try it and tell me the result by comentary in this build.Then i can change this build to a better one.

The items of this build may be a little stranger for you but believe, all they are very usefull and they conjunt make Corki a very dificult hero to kill, and very strong on battle.For me and not only for me Corki does not need any item of ap(trinity force),any item of mana regeneration(manamuke).
I respect other builds that like this items and i see players doing well with this items but if you compare the damage that my build have and the damage of these builds you may start to think like me.
As my build have no items of mana regen ,i use clarity to Corki and to me(i use skills all the time)mana never be a problem.
Attack speed is exelent for Corki but this build is focused in damage items, it have attack speed but it is focused in AD.Damage is the main source of Corki's skill,3 of 4 Corki's skill need damage.So damage items are more essentials for Corki than attack speed.
There are many othes Corki builds and a lot of then have good items ,so remember this is a farming and carry Corki build and some items of this build are farming ones and some carry ones(damage,life,armor pen,attack speed,armor),so this ain't a comum Corki build.
I know a lot of changes that could be made in this items build but if i do i will be doing a copy
of a already existent build.I accept any sugestion if it is correct for me.

Some peoples start with phosphorus bomb,i like to start with valkyrie because if the others guys try to gank you in begging you will be ready to escape.When you learn phosphorus bomb and valkyrie you will be lvl 3 until lvl 6 only upgrade gatling gun(Corki main source of damage in this build). At lvl 6 upgrade your ult for the first time.Then upgrade gatling gun always you can if you can't, upgrade phosphoros bomb(only do that if you can't upgrade your ult).When phosphorus bomb and gatling gun is at lvl 5 start to upgrade valkyrie if you can't upgrade your ult of course.
It's very easy to learn and and simple to make if you already don't know you will learn it easy.

I use:
Clarity:I use this one as i say before for farming and eventualy for fight.
Ghost : I use this one for running and for kill enemy in running.Good with ult to last hit and gatling gun to stop enemy running with slow(frozen mallet).
Good for Corki too:
Flash:This one is fantastic for Corki last hit enemys.I like more of ghost but if you want it its a good choice.
Ignite:Good choice for Corki too but i like more the others that i have choice.
Teleport:Teleport also good for him but his cooldown is to high.
No good for Corki:
Exhaust:Frozen mallet already gives slow and exhaust also decrease the damage to the slowed person.No reason for use.
Revive:I belive that i need no explanation about this one.

There others good spells for Corki but this ones i recomend.

When I was making this build,I take care about somethings that others builds doesn't like a little bit of armor,dodge chance and armor penetration.I know that may be a strange build but if you play with her of the right way you may be a match winner.So I make some coments about my build diference between the others and what this reflecs in gameplay.
1.This build have armor in 2 of his items what makes a Corki more stronger in battles and less vulnerable target in teamfights.
2.Dogde chance can be a very usual atribute when used with Corki 1 skill that makes oponents miss they attacks(35% chance)and dogde chance also makes Corki stronger on 1x1 battles.The boots + the runes + the masteries=30% dogde chance(first skill miss chance not on count).
3.Armor penetration for Corki is a very important atribute agaisnt almost all characters on the game expecial the tank one's.As you know Corki is a very damage character and armor penetration complete his ability to damage.
4.Some peoples may asking why clarity and I say:farming.Farming is one of the best Corki abilitis and if use a lot of skills on minions you may stay out of mana,but with clarity this not happens.Madred also gives you critical against minons.
5.This is one of the most complete builds for Corki it have:health,armor,dogde,atack speed,critical strike,armor penetration,and a lot of damage.BUT YOU ONLY WILL GET IT IF YOU FARMING.
6.There is many items that could be change.For example:Black Cleaver in place of Last Whisper,boots of attack speed for boots of armor(I dont recomend),The Brutalizer in place of Last Whisper(I dont recomend), and lot others.